cp's OEIS Frontend

A050334 Number of ordered factorizations of n into numbers with an odd number of prime divisors (prime factors counted with multiplicity).

%I A050334 #18 Jan 03 2021 15:56:33
%S A050334 1,1,1,1,1,2,1,2,1,2,1,4,1,2,2,3,1,4,1,4,2,2,1,8,1,2,2,4,1,7,1,5,2,2,
%T A050334 2,10,1,2,2,8,1,7,1,4,4,2,1,15,1,4,2,4,1,8,2,8,2,2,1,18,1,2,4,8,2,7,1,
%U A050334 4,2,7,1,23,1,2,4,4,2,7,1,15,3,2,1,18,2,2,2,8,1,18,2,4,2,2,2,28,1,4,4
%N A050334 Number of ordered factorizations of n into numbers with an odd number of prime divisors (prime factors counted with multiplicity).
%C A050334 a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).
%H A050334 R. J. Mathar, <a href="/A050334/b050334.txt">Table of n, a(n) for n = 1..10000</a>
%F A050334 Dirichlet g.f.: 1/(1-B(s)) where B(s) is D.g.f. of characteristic function of A026424 (essentially A066829).
%F A050334 a(p^k) = A000045(k).
%F A050334 a(A002110(k)) = A006154(k).
%F A050334 a(n) = A050335(A101296(n)). - _R. J. Mathar_, May 26 2017
%e A050334 From _R. J. Mathar_, May 25 2017: (Start)
%e A050334 a(p) = 1: factorizations p.
%e A050334 a(p^2) = 1: factorizations p*p.
%e A050334 a(p^3) = 2: factorizations p^3, p*p*p.
%e A050334 a(p^4) = 3: factorizations p^3*p, p*p^3, p*p*p*p.
%e A050334 a(p^5) = 5: factorizations p^5, p^3*p*p, p*p^3*p, p*p*p^3, p*p*p*p*p.
%e A050334 a(p*q) = 2: factorizations p*q, q*p. (End)
%p A050334 read(transforms):
%p A050334 A066829m := proc(n)
%p A050334         if n = 1 or isA026424(n) then
%p A050334                 1;
%p A050334         else
%p A050334                 0;
%p A050334         end if;
%p A050334 end proc:
%p A050334 [1,seq(-A066829m(n),n=2..10000)] ;
%p A050334 DIRICHLETi(%) ; # _R. J. Mathar_, May 25 2017
%Y A050334 Cf. A002033, A026424, A050333.
%K A050334 nonn
%O A050334 1,6
%A A050334 _Christian G. Bower_, Oct 15 1999